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comment Are natural integers intuitive?
You may want to read about ultrafinitism, but it sounds like that's not what you're getting at.
Mar
31
comment Can $[0,1]$ be partitioned with the following property?
I may be misreading something. If the measure of A is less than 1/2, isn't there no hope?
Mar
30
comment Infinite number of Derivatives
@user31415 My confusion was because my comment was about a different theorem. But as I said above, I now notice it's not needed for that other theorem either. I thank you for pointing that out.
Mar
30
comment Infinite number of Derivatives
@user31415 I didn't post an answer, nor did I use the word smooth. But nonetheless you're right. I think the proof I linked to doesn't make use of infinite differentiability in any way.
Mar
30
comment Infinite number of Derivatives
@user31415 I don't think you meant to @ me, but I agree.
Mar
29
comment Infinite number of Derivatives
And it's a bit less easy to prove that any function which is infinitely differentiable and has some derivative equal to zero at each point is still a polynomial. See the MathOverflow question
Mar
29
comment What's the difference between hyperreal and surreal numbers?
Things get a little hairy when dealing with class-sized hyperreal fields, but Keisler is quoted with a precise statement of "the surreals are the biggest version of the hyperreals" and proof sketch in Ehrlich's "The absolute arithmetic continuum and the unification of all numbers great and small"
Mar
29
comment Numerical system that includes the limit targets such as $0^+$, $0^-$, $1^+$ etc
You have listed the "limit targets", and I don't see why you want numerical operations on them, nor do I have much of an intuition about how they should be defined. Why must it be that $1^+=1+0^+$? As an aside (to complicate matters), a good way of thinking about limits is in terms of limsup and liminf from the left and right. Then at each real number $a$, a function defined on, say, $(a-h,a+h)$ has four "limits", which are extended reals.
Mar
28
comment proving that a function has a third derivative
The definition of $h(x)$ really doesn't say anything about $f$. The problem probably meant to say, at the very least, that $f$ was thrice differentiable, as otherwise the definition of $h$ may be invalid.
Mar
27
comment Types of definitions
Compare "A field is a field of reals if it's dedekind complete." to "A field is a field of reals if it's isomorphic as an ordered field to the following: [dedekind cuts construction]". I think this is close to what Bitwise is getting at, but the distinction doesn't hold mathematical significance, to my knowledge.
Mar
27
comment How should one depict a sign graph/chart with only 1 critical point and one interval?
You could also show the whole real line and write "not defined". As long as it's clear what you're trying to represent, it should be fine.
Mar
26
comment Why hyperreal numbers are built so complicatedly?
@Anixx, Those expressions needn't even make sense for arbitrary real functions f. I can't explain this issue in a comment, but maybe I or someone else will write a new long answer to this question that clears this stuff up.
Mar
26
comment Why hyperreal numbers are built so complicatedly?
@Anixx For one thing, you can't define differentiability properly for all real functions without the transfer principle of the hyperreals. I suppose "is this function differentiable at x=17?" Would be an unanswered question.
Mar
26
comment Why hyperreal numbers are built so complicatedly?
@Anixx, I think you misunderstood my meaning. The purpose of the hyperreals is to have a framework for derivatives and integrals that let's you answer all the same questions as regular calculus, but with infinitesimal numbers, rather than epsilon delta definitions.
Mar
26
comment Negative infinity to square equals positive infinity?
I think you want parentheses around that $-\infty$.
Mar
26
comment Describe the diffrence between the following two problems and give an example of a physical situation which may be modeled by each equation
The Heaviside distribution is discontinuous, and is a forcing function in this equation.
Mar
26
comment All Eigenvalues of the operator $L(v)= L^2(v).$
@TedMosby, Since the answer posted shows that it must be 0 or 1, your equation, which is also true, doesn't contradict that. -1 doesn't satisfy the quadratic so it's actually impossible, even though the cubic makes it a candidate.
Mar
26
comment Has anybody ever considered “full derivative”?
@Anixx If you add an infinitesimal to rationals, you get formal Laurent series with rational coefficients. The Levi-Civita field is different in that it allows real coefficients and rational powers of $\epsilon$. But to define your full derivative in cases that aren't polynomials, you need a method for defining things like $sin(\epsilon)$. Hyperreal fields make this work perfectly, but physics.umanitoba.ca/~khodr/Publications/… suggests that the Levi-Civita field is probably good enough, at least for analytic functions (I haven't thought about it much).
Mar
25
comment Has anybody ever considered “full derivative”?
I think you misunderstand the notation in that PDF. $No(\omega)$ is actually the dyadic rationals thanks to the tree rank (see Theorem 15). $ No(\omega_1) $ is a hyperreal system assuming CH, but that has a lot of different flavors of infinitesimals, not just what you can get with reals and $\omega $.
Mar
25
comment Has anybody ever considered “full derivative”?
A side comment that has no bearing on this question: @Anixx, you can take $No(\omega)$, but as I said in answer to math.stackexchange.com/questions/1193422/… that won't give you the hyperreals.